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Theory of Groups of Finite Order and the Burnside Problem

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Theory of Groups of Finite Order and the Burnside Problem Running head: WILLIAM BURNSIDE 1 William Burnside Theory of Groups of Finite Order and the Burnside Problem Maleah Mortorff A Senior Thesis submitted in partial fulfillment of the requirements for graduation in the Honors Program Liberty University Spring 2016 WILLIAM BURNSIDE 2 Acceptance of Senior Honors Thesis This Senior Honors Thesis is accepted in partial fulfillment of the requirements for graduation from the Honors Program of Liberty University. ______________________________ Timothy Sprano, Ph.D. Thesis Chair ______________________________ Evangelos Skoumbourdis, Ph.D. Committee Member ______________________________ Harvey Hartman, Th.D. Committee Member ______________________________ Brenda Ayres, Ph.D. Honors Director ______________________________ Date WILLIAM BURNSIDE 3 Abstract As one of the most influential founders of modern group theory, William Burnside and his work generated initial interest in the field of group theory. His book Theory of Groups of Finite Order was regarded for several decades as the standard measure for group research. Namely, the General Burnside Problem examines a finitely generated periodic group, questioning whether or not that group must be necessarily finite. Breakdowns in this general problem led to a definitive negative answer by Evgeny Golod and Tgor Shararevich in 1964, but paved the way for research into specific cases such as the prime exponent. This thesis will consider the background of Burnside’s mathematical expertise, as well as the general, bounded, and restricted cases of Burnside’s problem, concluding with a brief overview of his theorem and lemma. WILLIAM BURNSIDE 4 William Burnside: Theory of Groups of Finite Order and the Burnside Problem Influential as a founder of modern group theory, William Burnside generated the initial interest that brought group research to the forefront of mathematics. Publishing over one hundred and sixty papers, three books, and serving on countless committees and public offices, he was both a patient teacher and a ruthless researcher (Burnside, Neumann, & Tompson, p. 15). Topics of interest to him included but were not limited to: elliptic functions, hydrodynamics, electromagnetic theory, differential geometry, projective geometry, statistical mechanics, general theory of functions and modular functions, group theory, and probability and statistics (p. 15). Burnside was characterized by an elegance and conciseness uncommon amongst his peers, and was known to be severe in his high standards yet fruitful in suggestion (Adian, p. 28). While not particularly fond of working with others in terms of group collaboration, on a few separate occasions he broke this trend for the love of his research (Adelmann & Gerbracht, p. 34). Although not always beneficial to his mathematical pursuits, the remains of the correspondence between Burnside and a few elect contemporaries give current readers a look into who he was both personally and academically. His work in finite group theory ignited a generation of interest in the field, and his work Theory of Groups of Finite Order was regarded for several decades as the foundational starting point for any work falling under the category groups. Namely, the General Burnside Problem examines a finitely generated, periodic group, and asks whether or not that group must necessarily be finite. While the general problem broke down into a negative answer, later many specific cases such as the prime exponent and others were proven to be true. Relevant topics to be considered in the study of Burnside WILLIAM BURNSIDE 5 include the General Burnside Problem, the Bounded Burnside Problem, and the restricted case of Burnside’s Problem, Burnside’s theorem, Burnside’s lemma, Burnside’s ring, and other considerable influences he made on the mathematical community in general. Burnside’s Background On July 2, 1852, William Burnside was born in London to William Burnside and Emma (Knight) Burnside (Forsyth, p. 64). At the age of six the younger William and his family unfortunately suffered the loss of William senior to apoplexy, and while previously financially stable, the death left them in monetary turmoil. Despite this lack of familial funding, Burnside’s mother was able to petition for a scholarship for him, and her son was able to attain his early schooling at Christ’s Hospital School, where he achieved the highest achievement in the mathematical school (Burnside, Neumann, & Tompson, p. 89). Upon graduation in 1871, he then attended St. John’s College in Cambridge, and was considered the best man of his year (p. 15). Late in his second year of undergraduate work, he moved to Pembroke College, graduating in 1875 (p. 15). He was known as an expert oarsman while there, and was extremely fond of and talented at the sport due to his light weight, spare build, and powerful endurance capabilities (Forsyth, p. 64). After graduation, this love for rowing then turned into a zest and love for fishing, a passion he carried for the rest of his life. Upon graduation he received first prize in Smith’s competition examination, a prestigious competition for mathematicians which was considered a high honor, and was elected a Fellow of Pembroke, a title which he maintained from 1875 until 1886 (Forsyth, p. 64). He spent the next ten years of his life coaching both mathematics and WILLIAM BURNSIDE 6 rowing, both at Pembroke as well as Emmanuel (1876) and Kings (1877) (Burnside, Neumann, & Tompson, p. 15). He was published for the first time in 1883 (2004). In 1885, Burnside was offered a position as a professor at Royal Naval College in Greenwich which he accepted, choosing to spend the rest of his teaching career at that location (Burnside, Neumann, & Tompson, p. 95). Several years later, he was offered an administrative position back at his alma mater but refused, perhaps partially due to the relaxed lifestyle his current teaching allowed him (p. 97). His time was devoted to teaching and training naval officers, particularly those advanced in areas of kinematics, kinetics, and hydrodynamics. Said to have a patient, stimulating teaching style, Burnside was well loved and admired by his students, although at times he could be said to be a bit severe in his grading (Forsyth, p. 72). At this point in his career his publications began to take off, and by 1887 he averaged about four papers a year (Burnside, Neumann, & Tompson, p. 15). In terms of family life, Burnside married Alexandrina Urquhart in 1886, and together they were the parents of two sons and three daughters (Forsyth, p. 73). He received his first major honor upon election as a fellow of the Royal Society in 1893 (Burnside, Neumann, & Tompson, p. 15). In this same year, he also published his first paper on group theory (Burnside & Panton, p. 50). Of special interest in this paper was a proof that, excepting (A5), no finite simple group exists whose order is the product of four primes (Burnside, Neumann, & Tompson, p. 31). His first paper was later followed by the publication of his first book, Theory of Groups of Finite Order. In the year 1900 he received what he considered his most meaningful award of honorary status as a Fellow of Pembroke (p. 91). WILLIAM BURNSIDE 7 The Burnside Problem, one of the propositions for which he is best known, was theorized for the first time in 1902 (Burnside, Neumann, & Tompson, p. 15). This initial proposition was followed closely in 1904 by his proposal of the (paqb) theorem (p. 15). In 1906, with what has been deemed “grave and characteristic reluctance,” (p. 15), Burnside was elected as President of the London Mathematical Society. He served in this role for two years, in addition to being a member of the council from 1899 until 1917 (p. 16). He republished a second edition of Theory of Groups of Finite Order in 1911, a version which contained five additional chapters and over fifty percent more content than the original (p. 16). Burnside’s retirement came in 1919, at age sixty seven, upon which time he moved to the country where he could maintain a more leisurely existence (Forsyth, p. 73). He continued his research work until his death, especially in his final years of life delving into the world of probability and statistics. William Burnside died of cerebral hemorrhage on August 21, 1927, at the age of seventy five (Burnside, Neumann, & Tompson, p. 106). Posthumously, his work Theory of Probability was later published (p. 15). Personally, Burnside was a well-liked, if sometimes severe individual (Forsyth, p. 75). He was known to set high standards both for himself and for his colleagues and students, but was able to maintain sympathy even in this critique (Burnside, Neumann, & Tompson, p. 15). Burnside despised the pomp and circumstance that accompanied many prestigious awards and positions, and preferred to work individually. While many contemporary mathematicians of his day collaborated frequently, Burnside preferred to work privately, and to separate himself from mathematical controversy. According to The WILLIAM BURNSIDE 8 Collected Papers of William Burnside (p. 15), “Although from 1895 onwards Burnside seems to have kept himself well informed about the published literature of group theory, he does not appear to have had extensive direct contacts with other mathematicians interested in the subject.” Some notable exceptions appear, but as a general rule Burnside seemingly preferred to work alone. One of the primary aspects that set his work apart from many of his peers was the elegance of his writing style (p. 15). His mathematics were characterized by clear, precise thinking throughout the entirety of a developed issue, as well as a faculty of lucid expression throughout the argument (p. 23). As a rule, he was a very concise writer, a fact which set him apart both as a teacher and as a mathematician. Group Theory Foundation One of the most significant building blocks for Burnside’s prominence in the mathematical community was his foundational work in group theory (Forsyth, p.
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